For each n, let $U_n$ be Haar distributed on the group of $n\times n$ unitary matrices. Let $x_{n,1},\dots ,x_{n,m}$ denote orthogonal nonrandom unit vectors in ${ℂ}^n$ and let $u_{n,k}={(u_k^1,\dots ,u_k^n)}^{\ast }=U_n^{\ast }{x}_{n,k}$, $k=1,\dots ,m$. Define the following functions on $[0,1]$: $X_n^{k,k}(t)=\sqrt{n}{\sum }_{i=1}^{[nt]}(|u_k^i{|}^2-\frac{1}{n})$, $X_n^{k,k^{\prime }}(t)=\sqrt{2n}{\sum }_{i=1}^{[nt]}{ū}_k^i{u}_{k^{\prime }}^i$, $k<k^{\prime }$. Then it is proven that $X_n^{k,k},\Re X_n^{k,k^{\prime }}$, $\Im X_n^{k,k^{\prime }}$, considered as random processes in $D[0,1]$, converge weakly, as $n\to \infty$, to $m^2$ independent copies of Brownian bridge. The same result holds for the $m(m+1)/2$ processes in the real case, where $O_n$ is real orthogonal Haar distributed and $x_{n,i}\in {ℝ}^n$, with $\sqrt{n}$ in $X_n^{k,k}$ and $\sqrt{2n}$ in $X_n^{k,k^{\prime }}$ replaced with $\sqrt{\frac{n}{2}}$ and $\sqrt{n}$, respectively. This latter result will be shown to hold for the matrix of eigenvectors of $M_n=(1/s)V_n{V}_n^T$ where $V_n$ is $n\times s$ consisting of the entries of $\{v_{ij}\},i,j=1,2,\dots$, i.i.d. standardized and symmetrically distributed, with each $x_{n,i}=\{\pm 1/\sqrt{n},\dots ,\pm 1/\sqrt{n}\}$ and $n/s\to y>0$ as $n\to \infty$. This result extends the result in [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied to the detection problem in sampling random vectors mostly made of noise and detecting whether the sample includes a nonrandom vector. The matrix $B_n=𝜃v_n{v}_n^{\ast }+S_n$ is studied where $S_n$ is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or $S_n=M_n$, $𝜃>0$ nonrandom and $v_n$ is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to $v_n$ with the eigenvector associated with the largest eigenvalue of $B_n.$

对于每个n，让${ü}_n$是 Haar 分布在组上$n\times n$酉矩阵。让$X_{n,1},\dots ,X_{n,米}$表示正交非随机单位向量${ℂ}^n$然后让${你}_{n,ķ}={({你}_{ķ}^1,\dots ,{你}_{ķ}^n)}^{*}={ü}_n^{*}{X}_{n,ķ}$,$ķ=1,\dots ,米$. 定义以下函数$[0,1]$：$X_n^{ķ,ķ}(吨)=\sqrt{n}{\sum }_{\mathrm{一世}=1}^{[n吨]}(|{你}_{ķ}^{\mathrm{一世}}{|}^2-\frac{1}{n})$,$X_n^{ķ,{ķ}^{\text{'}}}(吨)=\sqrt{2n}{\sum }_{\mathrm{一世}=1}^{[n吨]}{ū}_{ķ}^{\mathrm{一世}}{你}_{{ķ}^{\text{'}}}^{\mathrm{一世}}$,$ķ<{ķ}^{\text{'}}$. 然后证明$X_n^{ķ,ķ},\Re X_n^{ķ,{ķ}^{\text{'}}}$,$\Im X_n^{ķ,{ķ}^{\text{'}}}$，被认为是随机过程$D[0,1]$, 弱收敛，如$n\to \infty$， 至${米}^2$布朗桥的独立副本。同样的结果也适用于$米(米+1)/2$真实案例中的过程，其中${○}_n$是实正交 Haar 分布且$X_{n,\mathrm{一世}}\in {ℝ}^n$， 和$\sqrt{n}$在$X_n^{ķ,ķ}$和$\sqrt{2n}$在$X_n^{ķ,{ķ}^{\text{'}}}$替换为$\sqrt{\frac{n}{2}}$和$\sqrt{n}$， 分别。后一个结果将被证明适用于特征向量矩阵${米}_n=(1/s){五}_n{五}_n^{吨}$在哪里${五}_n$是$n\times s$由以下条目组成$\{v_{\mathrm{一世}j}\},\mathrm{一世},j=1,2,\dots$, iid 标准化且对称分布，每个$X_{n,\mathrm{一世}}=\{\pm 1/\sqrt{n},\dots ,\pm 1/\sqrt{n}\}$和$n/s\to \mathrm{是的}>0$作为$n\to \infty$. 该结果扩展了 [JW Silverstein, Ann. 概率。 18 (1990) 1174–1194]。这些结果适用于采样主要由噪声构成的随机向量和检测样本是否包含非随机向量的检测问题。矩阵${乙}_n=𝜃v_n{v}_n^{*}+{\mathrm{小号}}_n$在哪里研究${\mathrm{小号}}_n$是 Hermitian 或对称非负定，其特征向量矩阵为 Haar 分布，或${\mathrm{小号}}_n={米}_n$,$𝜃>0$非随机和$v_n$是一个非随机单位向量。结果是根据正交于向量的内积的分布行为得出的$v_n$与最大特征值相关的特征向量${乙}_n.$